Chapter 12.5, Problem 17E

Investigation Consider the graph of the vector-valued function $r(t)=ti+(4-t^2)j+t^{3}k$ on the interval [0, 2].

(a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints.

(b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the vectors r(0), r(0.5), r(l), r(1.5), and r(2).

(c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b).

(d) Use the integration capabilities of a graphing utility to approximate the length of the curve. Compare this result with the answers in parts (a) and (b).